Chicago to Poland - Cheap

Looking to travel to Poland cheap in June - June 1-June 16 to be exact. LOT is obvious best choice for direct flight at $1,370 (even compared to those with a stop over), but I'm hoping to save a couple hundred bucks by getting a good deal to another European airport/Country and then continuing my travels with the likes of RyanAir or German Wings. Question is, where to fly to to get the best deal? Does anyone have recommendations on websites or airlines?